A Jewel At The Heart Of Quantum Physics
Quanta MagazineA Jewel at the Heart of Quantum PhysicsBy Natalie WolchoverIllustration by Andy GilmoreArtist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel inhigher dimensions.Physicists have discovered a jewel-like geometric object that dramatically simplifies calculations ofparticle interactions and challenges the notion that space and time are fundamental components ofreality.“This is completely new and very much simpler than anything that has been done before,” saidAndrew Hodges, a mathematical physicist at Oxford University who has been following the work.The revelation that particle interactions, the most basic events in nature, may be consequences ofgeometry significantly advances a decades-long effort to reformulate quantum field theory, the bodyof laws describing elementary particles and their interactions. Interactions that were 130917/September 17, 2013
Quanta Magazinecalculated with mathematical formulas thousands of terms long can now be described by computingthe volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-termexpression.“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at HarvardUniversity and one of the researchers who developed the new idea. “You can easily do, on paper,computations that were infeasible even with a computer before.”The new geometric version of quantum field theory could also facilitate the search for a theory ofquantum gravity that would seamlessly connect the large- and small-scale pictures of the universe.Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run upagainst nonsensical infinities and deep paradoxes. The amplituhedron, or a similar geometric object,could help by removing two deeply rooted principles of physics: locality and unitarity.“Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professorof physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the newwork, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”Locality is the notion that particles can interact only from adjoining positions in space and time. Andunitarity holds that the probabilities of all possible outcomes of a quantum mechanical interactionmust add up to one. The concepts are the central pillars of quantum field theory in its original form,but in certain situations involving gravity, both break down, suggesting neither is a fundamentalaspect of nature.In keeping with this idea, the new geometric approach to particle interactions removes locality andunitarity from its starting assumptions. The amplituhedron is not built out of space-time andprobabilities; these properties merely arise as consequences of the jewel’s geometry. The usualpicture of space and time, and particles moving around in them, is a construct.“It’s a better formulation that makes you think about everything in a completely different way,” saidDavid Skinner, a theoretical physicist at Cambridge University.The amplituhedron itself does not describe gravity. But Arkani-Hamed and his collaborators thinkthere might be a related geometric object that does. Its properties would make it clear why particlesappear to exist, and why they appear to move in three dimensions of space and to change over time.Because “we know that ultimately, we need to find a theory that doesn’t have” unitarity and locality,Bourjaily said, “it’s a starting point to ultimately describing a quantum theory of gravity.”Clunky MachineryThe amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in itsvolume are the most basic features of reality that can be calculated, “scattering amplitudes,” whichrepresent the likelihood that a certain set of particles will turn into certain other particles uponcolliding. These numbers are what particle physicists calculate and test to high precision at particleaccelerators like the Large Hadron Collider in 20130917/September 17, 2013
Quanta MagazineThe iconic 20th century physicistRichard Feynman invented a method for calculating probabilities of particle interactions usingdepictions of all the different ways an interaction could occur. Examples of “Feynman diagrams”were included on a 2005 postage stamp honoring Feynman.The 60-year-old method for calculating scattering amplitudes — a major innovation at the time —was pioneered by the Nobel Prize-winning physicist Richard Feynman. He sketched line drawings ofall the ways a scattering process could occur and then summed the likelihoods of the differentdrawings. The simplest Feynman diagrams look like trees: The particles involved in a collision cometogether like roots, and the particles that result shoot out like branches. More complicated diagramshave loops, where colliding particles turn into unobservable “virtual particles” that interact witheach other before branching out as real final products. There are diagrams with one loop, two loops,three loops and so on — increasingly baroque iterations of the scattering process that contributeprogressively less to its total amplitude. Virtual particles are never observed in nature, but theywere considered mathematically necessary for unitarity — the requirement that probabilities sum toone.“The number of Feynman diagrams is so explosively large that even computations of really simpleprocesses weren’t done until the age of computers,” Bourjaily said. A seemingly simple event, suchas two subatomic particles called gluons colliding to produce four less energetic gluons (whichhappens billions of times a second during collisions at the Large Hadron Collider), involves 220diagrams, which collectively contribute thousands of terms to the calculation of the scatteringamplitude.In 1986, it became apparent that Feynman’s apparatus was a Rube Goldberg machine.To prepare for the construction of the Superconducting Super Collider in Texas (a project that waslater canceled), theorists wanted to calculate the scattering amplitudes of known particleinteractions to establish a background against which interesting or exotic signals would stand out.But even 2-gluon to 4-gluon processes were so complex, a group of physicists had written two yearsearlier, “that they may not be evaluated in the foreseeable future.”Stephen Parke and Tomasz Taylor, theorists at Fermi National Accelerator Laboratory in Illinois,took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula, which a1980s supercomputer could handle. Then, based on a pattern they observed in the scatteringamplitudes of other gluon interactions, Parke and Taylor guessed a simple one-term expression September 17, 2013
Quanta Magazinethe amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, thetraditional machinery of quantum field theory, involving hundreds of Feynman diagrams worththousands of mathematical terms, was obfuscating something much simpler. As Bourjaily put it:“Why are you summing up millions of things when the answer is just one function?”“We knew at the time that we had an important result,” Parke said. “We knew it instantly. But whatto do with it?”The AmplituhedronTwistor diagrams depicting aninteraction between six gluons, in the cases where two (left) and four (right) of the particles havenegative helicity, a property similar to spin. The diagrams can be used to derive a simple formula forthe 6-gluon scattering amplitude.The message of Parke and Taylor’s single-term result took decades to interpret. “That one-term,beautiful little function was like a beacon for the next 30 years,” Bourjaily said. It “really started thisrevolution.”In the mid-2000s, more patterns emerged in the scattering amplitudes of particle interactions,repeatedly hinting at an underlying, coherent mathematical structure behind quantum field theory.Most important was a set of formulas called the BCFW recursion relations, named for Ruth Britto,Freddy Cachazo, Bo Feng and Edward Witten. Instead of describing scattering processes in terms offamiliar variables like position and time and depicting them in thousands of Feynman diagrams, theBCFW relations are best couched in terms of strange variables called “twistors,” and particleinteractions can be captured in a handful of associated twistor diagrams. The relations gained rapidadoption as tools for computing scattering amplitudes relevant to experiments, such as collisions atthe Large Hadron Collider. But their simplicity was mysterious.“The terms in these BCFW relations were coming from a different world, and we wanted tounderstand what that world was,” Arkani-Hamed said. “That’s what drew me into the subject fiveyears ago.”With the help of leading mathematicians such as Pierre Deligne, Arkani-Hamed and his collaboratorsdiscovered that the recursion relations and associated twistor diagrams corresponded to a wellknown geometric object. In fact, as detailed in a paper posted to arXiv.org in December by ArkaniHamed, Bourjaily, Cachazo, Alexander Goncharov, Alexander Postnikov and Jaroslav Trnka, thetwistor diagrams gave instructions for calculating the volume of pieces of this object, called September 17, 2013
Quanta Magazinepositive Grassmannian.Asketch of the amplituhedron representing an 8-gluon particle interaction. Using Feynman diagrams,the same calculation would take roughly 500 pages of algebra.Named for Hermann Grassmann, a 19th-century German linguist and mathematician who studied itsproperties, “the positive Grassmannian is the slightly more grown-up cousin of the inside of atriangle,” Arkani-Hamed explained. Just as the inside of a triangle is a region in a two-dimensionalspace bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in anN-dimensional space bounded by intersecting planes. (N is the number of particles involved in ascattering process.)It was a geometric representation of real particle data, such as the likelihood that two collidinggluons will turn into four gluons. But something was still 0917/September 17, 2013
Quanta MagazineThe physicists hoped that the amplitude of a scattering process would emerge purely and inevitablyfrom geometry, but locality and unitarity were dictating which pieces of the positive Grassmannianto add together to get it. They wondered whether the amplitude was “the answer to some particularmathematical question,” said Trnka, a post-doctoral researcher at the California Institute ofTechnology. “And it is,” he said.Arkani-Hamed and Trnka discovered that the scattering amplitude equals the volume of a brand-newmathematical object — the amplituhedron. The details of a particular scattering process dictate thedimensionality and facets of the corresponding amplituhedron. The pieces of the positiveGrassmannian that were being calculated with twistor diagrams and then added together by handwere building blocks that fit together inside this jewel, just as triangles fit together to form apolygon.Like the twistor diagrams, the Feynman diagrams are another way of computing the volume of theamplituhedron piece by piece, but they are much less efficient. “They are local and unitary in spacetime, but they are not necessarily very convenient or well-adapted to the shape of this jewel itself,”Skinner said. “Using Feynman diagrams is like taking a Ming vase and smashing it on the floor.”Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron directly insome cases, without using twistor diagrams to compute the volumes of its pieces. They have alsofound a “master amplituhedron” with an infinite number of facets, analogous to a circle in 2-D,which has an infinite number of sides. Its volume represents, in theory, the total amplitude of allphysical processes. Lower-dimensional amplituhedra, which correspond to interactions betweenfinite numbers of particles, live on the faces of this master structure.“They are very powerful calculational techniques, but they are also incredibly suggestive,” Skinnersaid. “They suggest that thinking in terms of space-time was not the right way of going about this.”Quest for Quantum GravityThe seemingly irreconcilable conflict between gravity and quantum field theory enters crisis mode inblack holes. Black holes pack a huge amount of mass into an extremely small space, making gravitya major player at the quantum scale, where it can usually be ignored. Inevitably, either locality orunitarity is the source of the conflict.“We have indications that both ideas have got to go,” Arkani-Hamed said. “They can’t befundamental features of the next description,” such as a theory of quantum gravity.String theory, a framework that treats particles as invisibly small, vibrating strings, is one candidatefor a theory of quantum gravity that seems to hold up in black hole situations, but its relationship toreality is unproven — or at least confusing. Recently, a strange duality has been found betweenstring theory and quantum field theory, indicating that the former (which includes gravity) ismathematically equivalent to the latter (which does not) when the two theories describe the sameevent as if it is taking place in different numbers of dimensions. No one knows quite what to make ofthis discovery. But the new amplituhedron research suggests space-time, and therefore dimensions,may be illusory anyway.“We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics,”Arkani-Hamed said. “We have to learn new ways of talking about it. This work is a baby step in thatdirection.”Even without unitarity and locality, the amplituhedron formulation of quantum field theory does September 17, 2013
Quanta Magazineyet incorporate gravity. But researchers are working on it. They say scattering processes thatinclude gravity particles may be possible to describe with the amplituhedron, or with a similargeometric object. “It might be closely related but slightly different and harder to find,” Skinner said.Physicists must also prove that the new geometric formulation applies to the exact particles that areknown to exist in the universe, rather than to the idealized quantum field theory they used todevelop it, called maximally supersymmetric Yang-Mills theory. This model, which includes a“superpartner” particle for every known particle and treats space-time as flat, “just happens to bethe simplest test case for these new tools,” Bourjaily said. “The way to generalize these new tools to[other] theories is understood.”Nima Arkani-Hamed, aprofessor at the Institute for Advanced Study, and his former student and co-author Jaroslav Trnka,who finished his Ph.D. at Princeton University in July and is now a post-doctoral researcher at theCalifornia Institute of Technology.Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery ofthe amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving upspace and time as fundamental constituents of nature and figuring out how the Big Bang andcosmological evolution of the universe arose out of pure geometry.“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s notfrom the object changing. The object is basically timeless.”While more work is needed, many theoretical physicists are paying close attention to the new ideas.The work is “very unexpected from several points of view,” said Witten, a theoretical physicist at theInstitute for Advanced Study. “The field is still developing very fast, and it is difficult to guess whatwill happen or what the lessons will turn out to be.”Note: This article was updated on December 10, 2013, to include a link to the first in a series ofpapers on the amplituhedron.This article was reprinted on 130917/September 17, 2013
The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes.
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Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
